| The hydrodynamics of Stokes flows are very important in the study of rheology, lubrication theory, and micro-organism locomotion and other biophysical systems. In particular, the dynamics of elastic filaments plays an important role in the theories of polymeric liquids, DNA and bacterial supercoiling, and pattern formation in liquid crystal systems. We are particularly interested in the dynamics of a growing, elastic filament immersed in a Stokesian fluid, as a model for the pattern formation seen in experiments of phase transitions of some smectic-A liquid crystals. There, a filament in the smectic phase grows by the intake of material from the fluid, with the "splay" deformation of the liquid crystal filament giving it an elastic response.;Our model combines slender-body theory, and filament elasticity and tensile forces. A general slender-body description was developed by (among others) Keller & Rubinow (K-R, 1976) who used methods of matched asymptotics. Using this as the basis for a dynamical model of smectic filament, Shelley & Ueda found a high-wavenumber instability, and reformulated the asymptotic analysis to remove it. They performed 2-D simulations of the resulting pattern formation. In this thesis, we derive a new model for slender body hydrodynamics from a boundary integral representation, and which we prove is asymptotically equivalent to the original K-R model. This new model is more suitable for simulating multiple, interacting filaments. The numerical methods of Shelley & Ueda were restricted to 2-D, and here are extended to 3-D in a fashion that allows natural implicit treatment of high-order terms. We then simulate the interaction of two filaments in 3-D, where the nonlocal hydrodynamics leads to mutual avoidance. |
